Estimating the Local Radius of Convergence for Picard Iteration

The well known Ostrowski theorem [1] gives a sufficient condition (the spectral radius of the Jacobian of the iteration mapping in the fixed point to be less than 1) for the local convergence of Picard iteration. “However, no estimate for the size of an attraction ball is known” [2] (2009). The problem of estimating the local radius of convergence for different iterative methods was considered by numerous authors and several results were obtained particularly for Newton method and its variants. Nevertheless “... effective, computable estimates for convergence radii are rarely available” [3] (1975). A similar remark was made in a more recent paper [4] (2015): “The location of starting approximations, from which the iterative methods converge to a solution of the equation, is a difficult problem to solve”. It is worth noticing that the shape of the attraction basins is an unpredictable and sophisticated set, especially for high order methods, and therefore finding a good ball of convergence for these methods is indeed a difficult task. Among the oldest known results on this topic we could mention those given by Vertgeim, Rall, Rheinboldt, Traub and Wozniakowski, Deuflhard and Potra, Smale [3,5–9]. Relatively recent results were communicated by Argyros [10–12], Ferreira [13], Hernandez-Veron and Romero [4], Ren [14], Wang [15]. Deuflhard and Potra [5] proposed the following estimation for Newton method. Let F : D ⊂ X → Y be a nonlinear mapping, where X, Y are Banach spaces. Suppose that F is Fréchet differentiable on D, that F′(x) is invertible for each x ∈ D, and that ‖F′(x)−1(F′(x + s(y− x))− F′(x))(y− x)‖ ≤ sω‖y− x‖2,